On the $3$-coloring of planar graphs without cycles of length from $4$ to $6$

In $1976$, Steinberg conjectured that every planar graph without $4$- and $5$-cycles is $3$-colorable. This conjecture was proved false by Cohen-Addad et al in 2017. Erdős raised the following question: Is there an integer $k$ such that every planar graph without cycles of length from $4$ to $k$ is $3$-colorable? Borodin et al. proved that every planar graph without cycles of length from $4$ to $7$ is $3$-colorable [Planar graphs without cycles of length from $4$ to $7$ are $3$-colorable, J. Combin.Theory Ser. B, 93 (2005), 303–311]. However, the question whether every planar graph without cycles of length from $4$ to $6$ is $3$-colorable is not answered yet and full of challenges. A $7$-cycle is called a special $7$-cycle if it shares an edge with another $7$-cycle or $9$-cycle. In this paper, we prove that every planar graph without cycles of length from $4$ to $6$ and without special $7$-cycles is $3$-colorable which is an improvement of Borodin's result.